By Robbins-Siegmund theorem (see [1]} or [2]}), we immediately obtain \(\sum _t \beta _t\Vert \nabla \mathcal {L}_t\Vert ^2<\infty \) .
Since \(\sum _t \beta _t = \infty \) , we have \(\liminf _{t\rightarrow \infty }\Vert \nabla \mathcal {L}_t\Vert  = 0\) .
Furthermore, we observe that
\(\mathbb {E}{{\begin{pmatrix}{x}_{t+1} - {x}_t\\{\lambda }_{t+1}-{\lambda }_t\end{pmatrix}}^2} = \mathbb {E}{\mathbb {E}{{\begin{pmatrix}{x}_{t+1} - {x}_t\\{\lambda }_{t+1}-{\lambda }_t\end{pmatrix}}^2\mid \mathcal {F}_{t-1}}} \\ \stackrel{(\ref {equ:update})}{\le }\eta _t^2\mathbb {E}{\mathbb {E}{{\begin{pmatrix}{\bar{\Delta }}{x}_t\\{\bar{\Delta }}{\lambda }_t\end{pmatrix}}^2 \mid \mathcal {F}_{t-1}} }\stackrel{(\ref {pequ:7})}{\le } 16\Upsilon _K^2(\Upsilon _u^2\vee \Upsilon _m)\cdot \eta _t^2.\) 
